Download the black–scholes type financial models and the arbitrage

Revista de Matemática: Teorı́a y Aplicaciones 2007 14(1) : 1–6
cimpa – ucr – ccss
issn: 1409-2433
the black–scholes type financial models and
the arbitrage opportunities
Nikolay Sukhomlin∗
Recibido/Received: 15 Jul 2006 — Aceptado/Accepted: 31 Jan 2007
Abstract
By using the conservation laws concept, we study certain financial models similar to
the Black–Scholes model. We show that without complement limitations such models
can have two or more volatilities. This fact imposes several intrinsic limitations for
the dynamical system parameters in order to guarantee the correct definition.
Keywords: Black–Scholes model, volatility, laws of conservation.
Resumen
Usando el concepto de leyes de conservación, estudiamos ciertos modelos financieros
similares al modelo de Black–Scholes. Demostramos que sin limitaciones complementarias tales modelos pueden tener dos o más volatilidades. Este hecho impone varias
limitaciones intrı́nsecas para los parámetros de sistemas dinámicos con fines de garantizar la definición correcta de dichos sistemas.
Palabras clave: Modelo de Black–Scholes, volatilidad, leyes de conservación.
Mathematics Subject Classification: C12, C65.
1
Introduction
The classical solution to the Black–Scholes Equation has the following structure:
V (t, S) = SN (d1 ) − F (t)N (d2 )
∗
Department of Physics, the Autonomous University of Santo-Domingo (UASD), Ciudad Universitaria, Ap. 1000, Santo Domingo Dominican Republic, Phone: 809 686 31 19; and Department of Economics, the Pontifical Catholic University of Santo-Domingo (PUCMM), Ave. Abraham Lincoln, Ap.
2748, Santo Domingo, Dominican Republic; Phone 809 535 0111, ext 2144, Fax: 809 534 7060. E-Mail:
[email protected], [email protected]
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Rev.Mate.Teor.Aplic. (2007) 14(1)
with:
1
F (t) = Ke−r(T −t) , N (d) = √
2π
Z
d
e−u
2 /2
du,
−∞
ln(S/K)
− βτ,
τ
= d2 + τ
d2 =
d1
β=
then
S
√
1
r
− 2,τ = σ T − t
2 σ
(1)
dN (d1 )
dN (d2 )
= F (t)
d(d1 )
d(d2 )
(2)
where V is the value of the call option (theoretical call premium), S the current stock
price at the moment of time t, r is the risk-free interest rate, and σ is the volatility
(the last two parameters being supposed constant). The condition at the boundaries is:
V = max(S − K; 0). The constants K and T respectively represent the strike price and
the option expiration date.
This is a well known relation for the Black–Scholes classical model (see for example
[11]).
Let be the function:
ξ = V (2) − V (1) ; V (1) = ∂V /∂(ln S), V (2) = ∂ 2 V /∂(ln S)2 .
Using the cumulative normal distribution function property
d2 N (x)
dx2
(3)
= −x dNdx(x) we obtain:
(d1 )
ξ = τ1 S dN
d(d1 ) . Let the elasticity of the auxiliary function ξ be:
Eξ =
∂(ln ξ)
1
= − 2 ln(S/K) + β,
∂(ln S)
τ
hence
τ 2 (Eξ − β) − ln(S/K) = 0.
(4)
After the substitution of β and τ from expression (1) in expression (4), we express the
volatility as a function of market variables:
σ2 =
ln(K/S) − r(T − t)
(T − t)(Eξ − 1/2)
The verification of this formula by simulation (see [8], [9] and [10]) shows the 10−9 order
of precision. The expression 4 represents the conservation law correspondent to the classic
Black—Scholes solution (see [7]):
τ 2 (Eξ − β) + ln(S) = ln(K) = const.
Thus one expression that “conserves” its value during the dynamical system evolution is
constructed. It is possible to proceed in the same way with different expressions similar to
black–scholes type financial models and arbitrage opportunities
3
the Black—Scholes classic formula. For example, [1] deduce an expression for the Value of
a European Constant Underlying Elasticity in Strikes (CUES) call Option that generalizes
the Black—Scholes classic solution:
2
C(t, S) = e−ετ SN (d1 ) − F (t, S)N (d2 )
(5)
with:
K = q 1/(1−α) ,
F (t, S) = K(S/K)α exp
−αε − (α − 1)β + (α2 − 1)/2 τ 2 ,
ε = δ/σ 2 ,
ln(S/K)
d1 =
+ (1 − β − ε)τ,
τ
d2 = d1 + (α − 1)τ,
1
r
β =
− 2,
2√ σ
τ = σ T −t
and N (.) is a cumulative normal distribution function. By simple calculus we obtain a
formula equivalent to expression (2):
2
e−ετ S
dN (d1 )
dN (d2 )
= F (t, S)
.
d(d1 )
d(d2 )
(6)
Now the correspondent to characteristic ξ for the CUES problem is:
ξ = C (2) − (1 + α)C (1) + αC =
1 − α −ετ 2 dN (d1 )
S
e
.
τ
d(d1 )
(7)
Its elasticity is a linear function of ln(S/K) :
Eξ =
∂(ln ξ)
1
= − 2 ln(S/K) + β + ε.
∂(ln S)
τ
(8)
The conservation law for the value of an European CUES call Option appears in similar
form:
τ 2 (Eξ − β − ε) + ln(S) = ln(K) = const.
(9)
It is easy to verify that all formulas from expression (5) to expression (9), transform
into correspondent formulas for the classic Black–Scholes solution under the conditions:
α → 0, ε → 0. So all results established for the Black—Scholes model can be automatically
extended to the Blenman and Clark model (see [1]). Particularly the volatility for this
model is:
ln q/(1 − α) − ln(S) − r(T − t)
σ2 =
.
(T − t)(Eξ − ε − 1/2)
Using this expression it is possible to verify the reliability of the the model at [1] by
calculus with market data.
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Rev.Mate.Teor.Aplic. (2007) 14(1)
The situation is more complicated in the case of the Valuation of Defaultable Bonds
Using Signaling Process Model (see [2]) and its extension (see [5] and [6]). The solution
of a Black—Scholes type equation is:
P (t, S) = G(t)SN (d1 ) − F (t, S)N (d2 ) + a function of time
with:
α − σ 2 /2 = σ 2 ν,
m = −2(1 − β)ν,
√
τ = σ t,
ln(S/S0 )
d1 =
+ ντ,
τ
d2 = −d1 − mτ,
2
F (t, S) = G(S/S0 )m eβνmτ ,
α, β, σ, S0 = const, N (.) is a cumulative normal distribution function, and G(t) is certain
function on time. The parameter β characterizes the distinction between the Lo and Hui
model (β 6= 0) and the Cathcart and El-Jahel model (see [2]) (β = 0). By simple calculus
we obtain a formula equivalent to expression 2:
G
dN (d1 )
dN (d2 )
= F (t, S)
d(d1 )
d(d2 )
The equivalent of the auxiliary characteristic in expression 3 is:
ξ = P (2) − mP (1) = −
2d1 + mτ dN (d1 )
G
.
τ2
d(d1 )
In this case the elasticity Eξ is a rational function of ln(S/S0 ):
Eξ =
ξ (1)
2d2 + md1 − 2
=− 1
.
ξ
τ (2d1 + mτ )
Entonces:
ABτ 2 + (A + B) ln(S/S0 ) − 1 + τ −2 ln2 (S/S0 ) = 0
with:
A = Eξ + ν, B = βν.
We conclude two cases emerge:
1. If β = 0 (the Cathcart and El-Jahel model, see [2]) one expression for the volatility
exists:
σ 2 = t−1 ln2 (S/S0 [1 − A ln(S/S0 )]−1
with evident restriction for the signaling variable: A ln(S/S0 ) < 1.
black–scholes type financial models and arbitrage opportunities
5
2. If β 6= 0 (Lo and Hui model, see [5] and [6]) there are two expressions for the volatility
under the conditions AB > 0, 1 > (A + B) ln(S/S0 ):
q
1
2
2
2
(σ )1,2 =
1 − (A + B) ln(S/S0 ) ± (A − B) ln (S/S0 ) − 2(A + B) ln(S/S0 ) + 1 .
ABt
To obtain only one expression it is necessary to change the conditions, for example to
impose AB < 0.
We can conclude that without certain restrictions the existence of two volatilities is
intrinsic for the Lo and Hui Valuation of Defaultable Bonds Using Signaling Process Model
and it is not in the case of the Cathcart and El-Jahel model. Lo and Hui introduce the
parameter β with the goal to approach to real market data. With such introduction
appears the possibility of model’s indeterminism. Another example can be found in [3]
and [4]. The authors use the Black and Scholes option pricing technique to develop an
expression for the Pricing of Defaultable Bonds with Log-Normal Spread. By similar
calculus we obtain a cubic equation with three solutions under certain conditions:
1
3
[λ(T − t) ln(h/H)] σ + 1 − (Eξ − ) ln(h/H) (T − t)σ 2 − ln2 (h/H) − µ ln(h/H) = 0
2
λ, T, H, σ, µ = const.
We can conclude that without specific restrictions the existence of two or tree volatilities
is intrinsic for the Cané de Estrada and all model (see [3] and [4]).
2
Conclusion
We show that it is possible to express the volatility for the Black and Scholes classic model
in function of market data. It is true in framework to this model. One expression for the
volatility and one conservation law exist. We find the important role of the elasticity in
this model too.
For other models the situation can be complicated: our procedure suggests the conclusion that several volatilities and conservation laws can exist.
We studied two concrete cases and found that under certain conditions two and three
volatilities exist. Without specific restrictions the existence of several volatilities is intrinsic
for the models mentioned.
It is possible to consider that the existence of two or more volatilities is related to the
presence of arbitrage opportunities and that the restrictions to have only one volatility
are related to the arbitrage absence.
Acknowledgements
Author is grateful to SEESCyT foundation, to Mr. Roberto Reyna and to the UASD
Department of Physics for their cooperation. I am especially grateful to the Editor in
chief Dr. Javier Trejos of the Revista de Matemática: Teorı́a y Aplicaciones for his useful
observations. Also, the author expresses his gratitude to Dr. Alemán and to PUCMM
Department of Economics.
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Rev.Mate.Teor.Aplic. (2007) 14(1)
References
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